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Tesi etd-07142011-235540

Thesis type
Tesi di dottorato di ricerca
Shape optimization problems of higher codimension
Settore scientifico disciplinare
Corso di studi
tutor Buttazzo, Giuseppe
Parole chiave
  • p-Laplacian equation
  • Location problems.
  • Network problems
  • Gamma-convergence
  • Compliance functional
  • Asymptotic shapes
  • Stazionary configuration
Data inizio appello
Riassunto analitico
The field of shape optimization problems has received a lot of attention in recent<br>years, particularly in relation to a number of applications in physics and engineering<br>that require a focus on shapes instead of parameters or functions. In general for ap-<br>plications the aim is to deform and modify the admissible shapes in order to optimize<br>a given cost function. The fascinating feature is that the variables are shapes, i.e.,<br>domains of R^{d}, instead of functions. This choice often produces additional dicul-<br>ties for the existence of a classical solution (that is an optimizing domain) and the<br>introduction of suitable relaxed formulation of the problem is needed in order to get a<br>solution which is in this case a measure. However, we may obtain a classical solution by<br>imposing some geometrical constraint on the class of competing domains or requiring<br>the cost functional verifies some particular conditions. The shape optimization problem<br>is in general an optimization problem of the form<br>min\{F(\Omega): \Omega \in {\cal O} \};<br>where F is a given cost functional and {\cal O} a class of domains in R^{d}. They are many<br>books written on shape optimization problems. The thesis is organized as follows: the<br>first chapter is dedicated to a brief introduction and presentation of some examples.<br>In Academic examples, we present the isoperimetric problems, minimal and capillary<br>surface problems and the spectral optimization problems while in applied examples the<br>Newton&#39;s problem of optimal aerodinamical profile and optimal mixture of two con-<br>ductors are considered. The second chapter is concerned with some basics elements of<br>geometric measure theory that will be used in the sequel. After recalling some notions<br>of abstract measure theory, we deal with the Hausdorff measures which are important<br>for defining the notion of approximate tangent space. Finally we introduce the notion<br>of approximate tangent space to a measure and to a set and also some differential op-<br>erators like tangential differential, tangential gradient and tangential divergence. The<br>third chapter is devoted to the topologies on the set of domains in R^{d}. Three topologies<br>induced by convergence of domains are presented namely the convergence of charac-<br>teristic functions, the convergence in the sense of Hausdorff and the convergence in<br>the sense of compacts as well as the relationship between those different topologies. In<br>the fourth chapter we present a shape optimization problem governed by linear state<br>equations. After dealing with the continuity of the solution of the Laplacian problem<br>with respect to the domain variation (including counter-examples to the continuity and<br>the introduction to a new topology: the <br>gamma-convergence), we analyse the existence of<br>optimal shapes and the necessary condition of optimality in the case where an optimal<br>shape exists. The shape optimization problems governed by nonlinear state equations are treated in chapter ve. The plan of study is the same as in chapter four that<br>is continuity with respect to the domain variation of the solution of the p-Laplacian<br>problem (and more general operator in divergence form), the existence of optimal<br>shapes and the necessary condition of optimality in the case where an optimal shape<br>exists. The last chapter deals with asymptotical shapes. After recalling the notion<br>of Gamma-convergence, we study the asymptotic of the compliance functional in different<br>situations. First we study the asymptotic of an optimal p-compliance-networks which<br>is the compliance associated to p-Laplacian problem with control variables running in<br>the class of one dimensional closed connected sets with assigned length. We provide<br>also the connection with other asymptotic problems like the average distance problem.<br>The asymptotic of the p-compliance-location which deal with the compliance associ-<br>atied to the p-Laplacian problem with control variables running in the class of sets of<br>finite numbers of points, is deduced from the study of the asymptotic of p-compliance-<br>networks. Secondly we study the asymptotic of an optimal compliance-location. In<br>this case we deal with the compliance associated to the classical Laplacian problem<br>and the class of control variables is the class of identics n balls with radius depending<br>on n and with fixed capacity.<br>